Optimal. Leaf size=299 \[ -\frac {4 b x}{a^5}+\frac {b^2 \left (20 a^6-35 a^4 b^2+28 a^2 b^4-8 b^6\right ) \tanh ^{-1}\left (\frac {\sqrt {a-b} \tan \left (\frac {1}{2} (c+d x)\right )}{\sqrt {a+b}}\right )}{a^5 (a-b)^{7/2} (a+b)^{7/2} d}+\frac {\left (6 a^6-65 a^4 b^2+68 a^2 b^4-24 b^6\right ) \sin (c+d x)}{6 a^4 \left (a^2-b^2\right )^3 d}+\frac {b^2 \sin (c+d x)}{3 a \left (a^2-b^2\right ) d (a+b \sec (c+d x))^3}+\frac {b^2 \left (9 a^2-4 b^2\right ) \sin (c+d x)}{6 a^2 \left (a^2-b^2\right )^2 d (a+b \sec (c+d x))^2}+\frac {b^2 \left (12 a^4-11 a^2 b^2+4 b^4\right ) \sin (c+d x)}{2 a^3 \left (a^2-b^2\right )^3 d (a+b \sec (c+d x))} \]
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Rubi [A]
time = 0.73, antiderivative size = 299, normalized size of antiderivative = 1.00, number of steps
used = 8, number of rules used = 7, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.368, Rules used = {3932, 4185,
4189, 4004, 3916, 2738, 214} \begin {gather*} -\frac {4 b x}{a^5}+\frac {b^2 \left (9 a^2-4 b^2\right ) \sin (c+d x)}{6 a^2 d \left (a^2-b^2\right )^2 (a+b \sec (c+d x))^2}+\frac {b^2 \sin (c+d x)}{3 a d \left (a^2-b^2\right ) (a+b \sec (c+d x))^3}+\frac {\left (6 a^6-65 a^4 b^2+68 a^2 b^4-24 b^6\right ) \sin (c+d x)}{6 a^4 d \left (a^2-b^2\right )^3}+\frac {b^2 \left (12 a^4-11 a^2 b^2+4 b^4\right ) \sin (c+d x)}{2 a^3 d \left (a^2-b^2\right )^3 (a+b \sec (c+d x))}+\frac {b^2 \left (20 a^6-35 a^4 b^2+28 a^2 b^4-8 b^6\right ) \tanh ^{-1}\left (\frac {\sqrt {a-b} \tan \left (\frac {1}{2} (c+d x)\right )}{\sqrt {a+b}}\right )}{a^5 d (a-b)^{7/2} (a+b)^{7/2}} \end {gather*}
Antiderivative was successfully verified.
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Rule 214
Rule 2738
Rule 3916
Rule 3932
Rule 4004
Rule 4185
Rule 4189
Rubi steps
\begin {align*} \int \frac {\cos (c+d x)}{(a+b \sec (c+d x))^4} \, dx &=\frac {b^2 \sin (c+d x)}{3 a \left (a^2-b^2\right ) d (a+b \sec (c+d x))^3}-\frac {\int \frac {\cos (c+d x) \left (-3 a^2+4 b^2+3 a b \sec (c+d x)-3 b^2 \sec ^2(c+d x)\right )}{(a+b \sec (c+d x))^3} \, dx}{3 a \left (a^2-b^2\right )}\\ &=\frac {b^2 \sin (c+d x)}{3 a \left (a^2-b^2\right ) d (a+b \sec (c+d x))^3}+\frac {b^2 \left (9 a^2-4 b^2\right ) \sin (c+d x)}{6 a^2 \left (a^2-b^2\right )^2 d (a+b \sec (c+d x))^2}+\frac {\int \frac {\cos (c+d x) \left (6 a^4-23 a^2 b^2+12 b^4-2 a b \left (6 a^2-b^2\right ) \sec (c+d x)+2 b^2 \left (9 a^2-4 b^2\right ) \sec ^2(c+d x)\right )}{(a+b \sec (c+d x))^2} \, dx}{6 a^2 \left (a^2-b^2\right )^2}\\ &=\frac {b^2 \sin (c+d x)}{3 a \left (a^2-b^2\right ) d (a+b \sec (c+d x))^3}+\frac {b^2 \left (9 a^2-4 b^2\right ) \sin (c+d x)}{6 a^2 \left (a^2-b^2\right )^2 d (a+b \sec (c+d x))^2}+\frac {b^2 \left (12 a^4-11 a^2 b^2+4 b^4\right ) \sin (c+d x)}{2 a^3 \left (a^2-b^2\right )^3 d (a+b \sec (c+d x))}-\frac {\int \frac {\cos (c+d x) \left (-6 a^6+65 a^4 b^2-68 a^2 b^4+24 b^6+a b \left (18 a^4-7 a^2 b^2+4 b^4\right ) \sec (c+d x)-3 b^2 \left (12 a^4-11 a^2 b^2+4 b^4\right ) \sec ^2(c+d x)\right )}{a+b \sec (c+d x)} \, dx}{6 a^3 \left (a^2-b^2\right )^3}\\ &=\frac {\left (6 a^6-65 a^4 b^2+68 a^2 b^4-24 b^6\right ) \sin (c+d x)}{6 a^4 \left (a^2-b^2\right )^3 d}+\frac {b^2 \sin (c+d x)}{3 a \left (a^2-b^2\right ) d (a+b \sec (c+d x))^3}+\frac {b^2 \left (9 a^2-4 b^2\right ) \sin (c+d x)}{6 a^2 \left (a^2-b^2\right )^2 d (a+b \sec (c+d x))^2}+\frac {b^2 \left (12 a^4-11 a^2 b^2+4 b^4\right ) \sin (c+d x)}{2 a^3 \left (a^2-b^2\right )^3 d (a+b \sec (c+d x))}+\frac {\int \frac {-24 b \left (a^2-b^2\right )^3+3 a b^2 \left (12 a^4-11 a^2 b^2+4 b^4\right ) \sec (c+d x)}{a+b \sec (c+d x)} \, dx}{6 a^4 \left (a^2-b^2\right )^3}\\ &=-\frac {4 b x}{a^5}+\frac {\left (6 a^6-65 a^4 b^2+68 a^2 b^4-24 b^6\right ) \sin (c+d x)}{6 a^4 \left (a^2-b^2\right )^3 d}+\frac {b^2 \sin (c+d x)}{3 a \left (a^2-b^2\right ) d (a+b \sec (c+d x))^3}+\frac {b^2 \left (9 a^2-4 b^2\right ) \sin (c+d x)}{6 a^2 \left (a^2-b^2\right )^2 d (a+b \sec (c+d x))^2}+\frac {b^2 \left (12 a^4-11 a^2 b^2+4 b^4\right ) \sin (c+d x)}{2 a^3 \left (a^2-b^2\right )^3 d (a+b \sec (c+d x))}+\frac {\left (b^2 \left (20 a^6-35 a^4 b^2+28 a^2 b^4-8 b^6\right )\right ) \int \frac {\sec (c+d x)}{a+b \sec (c+d x)} \, dx}{2 a^5 \left (a^2-b^2\right )^3}\\ &=-\frac {4 b x}{a^5}+\frac {\left (6 a^6-65 a^4 b^2+68 a^2 b^4-24 b^6\right ) \sin (c+d x)}{6 a^4 \left (a^2-b^2\right )^3 d}+\frac {b^2 \sin (c+d x)}{3 a \left (a^2-b^2\right ) d (a+b \sec (c+d x))^3}+\frac {b^2 \left (9 a^2-4 b^2\right ) \sin (c+d x)}{6 a^2 \left (a^2-b^2\right )^2 d (a+b \sec (c+d x))^2}+\frac {b^2 \left (12 a^4-11 a^2 b^2+4 b^4\right ) \sin (c+d x)}{2 a^3 \left (a^2-b^2\right )^3 d (a+b \sec (c+d x))}+\frac {\left (b \left (20 a^6-35 a^4 b^2+28 a^2 b^4-8 b^6\right )\right ) \int \frac {1}{1+\frac {a \cos (c+d x)}{b}} \, dx}{2 a^5 \left (a^2-b^2\right )^3}\\ &=-\frac {4 b x}{a^5}+\frac {\left (6 a^6-65 a^4 b^2+68 a^2 b^4-24 b^6\right ) \sin (c+d x)}{6 a^4 \left (a^2-b^2\right )^3 d}+\frac {b^2 \sin (c+d x)}{3 a \left (a^2-b^2\right ) d (a+b \sec (c+d x))^3}+\frac {b^2 \left (9 a^2-4 b^2\right ) \sin (c+d x)}{6 a^2 \left (a^2-b^2\right )^2 d (a+b \sec (c+d x))^2}+\frac {b^2 \left (12 a^4-11 a^2 b^2+4 b^4\right ) \sin (c+d x)}{2 a^3 \left (a^2-b^2\right )^3 d (a+b \sec (c+d x))}+\frac {\left (b \left (20 a^6-35 a^4 b^2+28 a^2 b^4-8 b^6\right )\right ) \text {Subst}\left (\int \frac {1}{1+\frac {a}{b}+\left (1-\frac {a}{b}\right ) x^2} \, dx,x,\tan \left (\frac {1}{2} (c+d x)\right )\right )}{a^5 \left (a^2-b^2\right )^3 d}\\ &=-\frac {4 b x}{a^5}+\frac {b^2 \left (20 a^6-35 a^4 b^2+28 a^2 b^4-8 b^6\right ) \tanh ^{-1}\left (\frac {\sqrt {a-b} \tan \left (\frac {1}{2} (c+d x)\right )}{\sqrt {a+b}}\right )}{a^5 (a-b)^{7/2} (a+b)^{7/2} d}+\frac {\left (6 a^6-65 a^4 b^2+68 a^2 b^4-24 b^6\right ) \sin (c+d x)}{6 a^4 \left (a^2-b^2\right )^3 d}+\frac {b^2 \sin (c+d x)}{3 a \left (a^2-b^2\right ) d (a+b \sec (c+d x))^3}+\frac {b^2 \left (9 a^2-4 b^2\right ) \sin (c+d x)}{6 a^2 \left (a^2-b^2\right )^2 d (a+b \sec (c+d x))^2}+\frac {b^2 \left (12 a^4-11 a^2 b^2+4 b^4\right ) \sin (c+d x)}{2 a^3 \left (a^2-b^2\right )^3 d (a+b \sec (c+d x))}\\ \end {align*}
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Mathematica [A]
time = 1.68, size = 293, normalized size = 0.98 \begin {gather*} \frac {(b+a \cos (c+d x)) \sec ^4(c+d x) \left (-24 b (c+d x) (b+a \cos (c+d x))^3+\frac {6 b^2 \left (-20 a^6+35 a^4 b^2-28 a^2 b^4+8 b^6\right ) \tanh ^{-1}\left (\frac {(-a+b) \tan \left (\frac {1}{2} (c+d x)\right )}{\sqrt {a^2-b^2}}\right ) (b+a \cos (c+d x))^3}{\left (a^2-b^2\right )^{7/2}}+\frac {2 a b^5 \sin (c+d x)}{(-a+b) (a+b)}+\frac {5 a b^4 \left (3 a^2-2 b^2\right ) (b+a \cos (c+d x)) \sin (c+d x)}{(a-b)^2 (a+b)^2}-\frac {a b^3 \left (60 a^4-71 a^2 b^2+26 b^4\right ) (b+a \cos (c+d x))^2 \sin (c+d x)}{(a-b)^3 (a+b)^3}+6 a (b+a \cos (c+d x))^3 \sin (c+d x)\right )}{6 a^5 d (a+b \sec (c+d x))^4} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.32, size = 399, normalized size = 1.33 Too large to display
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: ValueError} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 773 vs.
\(2 (282) = 564\).
time = 3.31, size = 1603, normalized size = 5.36 \begin {gather*} \left [-\frac {48 \, {\left (a^{11} b - 4 \, a^{9} b^{3} + 6 \, a^{7} b^{5} - 4 \, a^{5} b^{7} + a^{3} b^{9}\right )} d x \cos \left (d x + c\right )^{3} + 144 \, {\left (a^{10} b^{2} - 4 \, a^{8} b^{4} + 6 \, a^{6} b^{6} - 4 \, a^{4} b^{8} + a^{2} b^{10}\right )} d x \cos \left (d x + c\right )^{2} + 144 \, {\left (a^{9} b^{3} - 4 \, a^{7} b^{5} + 6 \, a^{5} b^{7} - 4 \, a^{3} b^{9} + a b^{11}\right )} d x \cos \left (d x + c\right ) + 48 \, {\left (a^{8} b^{4} - 4 \, a^{6} b^{6} + 6 \, a^{4} b^{8} - 4 \, a^{2} b^{10} + b^{12}\right )} d x - 3 \, {\left (20 \, a^{6} b^{5} - 35 \, a^{4} b^{7} + 28 \, a^{2} b^{9} - 8 \, b^{11} + {\left (20 \, a^{9} b^{2} - 35 \, a^{7} b^{4} + 28 \, a^{5} b^{6} - 8 \, a^{3} b^{8}\right )} \cos \left (d x + c\right )^{3} + 3 \, {\left (20 \, a^{8} b^{3} - 35 \, a^{6} b^{5} + 28 \, a^{4} b^{7} - 8 \, a^{2} b^{9}\right )} \cos \left (d x + c\right )^{2} + 3 \, {\left (20 \, a^{7} b^{4} - 35 \, a^{5} b^{6} + 28 \, a^{3} b^{8} - 8 \, a b^{10}\right )} \cos \left (d x + c\right )\right )} \sqrt {a^{2} - b^{2}} \log \left (\frac {2 \, a b \cos \left (d x + c\right ) - {\left (a^{2} - 2 \, b^{2}\right )} \cos \left (d x + c\right )^{2} + 2 \, \sqrt {a^{2} - b^{2}} {\left (b \cos \left (d x + c\right ) + a\right )} \sin \left (d x + c\right ) + 2 \, a^{2} - b^{2}}{a^{2} \cos \left (d x + c\right )^{2} + 2 \, a b \cos \left (d x + c\right ) + b^{2}}\right ) - 2 \, {\left (6 \, a^{9} b^{3} - 71 \, a^{7} b^{5} + 133 \, a^{5} b^{7} - 92 \, a^{3} b^{9} + 24 \, a b^{11} + 6 \, {\left (a^{12} - 4 \, a^{10} b^{2} + 6 \, a^{8} b^{4} - 4 \, a^{6} b^{6} + a^{4} b^{8}\right )} \cos \left (d x + c\right )^{3} + {\left (18 \, a^{11} b - 132 \, a^{9} b^{3} + 239 \, a^{7} b^{5} - 169 \, a^{5} b^{7} + 44 \, a^{3} b^{9}\right )} \cos \left (d x + c\right )^{2} + 3 \, {\left (6 \, a^{10} b^{2} - 59 \, a^{8} b^{4} + 110 \, a^{6} b^{6} - 77 \, a^{4} b^{8} + 20 \, a^{2} b^{10}\right )} \cos \left (d x + c\right )\right )} \sin \left (d x + c\right )}{12 \, {\left ({\left (a^{16} - 4 \, a^{14} b^{2} + 6 \, a^{12} b^{4} - 4 \, a^{10} b^{6} + a^{8} b^{8}\right )} d \cos \left (d x + c\right )^{3} + 3 \, {\left (a^{15} b - 4 \, a^{13} b^{3} + 6 \, a^{11} b^{5} - 4 \, a^{9} b^{7} + a^{7} b^{9}\right )} d \cos \left (d x + c\right )^{2} + 3 \, {\left (a^{14} b^{2} - 4 \, a^{12} b^{4} + 6 \, a^{10} b^{6} - 4 \, a^{8} b^{8} + a^{6} b^{10}\right )} d \cos \left (d x + c\right ) + {\left (a^{13} b^{3} - 4 \, a^{11} b^{5} + 6 \, a^{9} b^{7} - 4 \, a^{7} b^{9} + a^{5} b^{11}\right )} d\right )}}, -\frac {24 \, {\left (a^{11} b - 4 \, a^{9} b^{3} + 6 \, a^{7} b^{5} - 4 \, a^{5} b^{7} + a^{3} b^{9}\right )} d x \cos \left (d x + c\right )^{3} + 72 \, {\left (a^{10} b^{2} - 4 \, a^{8} b^{4} + 6 \, a^{6} b^{6} - 4 \, a^{4} b^{8} + a^{2} b^{10}\right )} d x \cos \left (d x + c\right )^{2} + 72 \, {\left (a^{9} b^{3} - 4 \, a^{7} b^{5} + 6 \, a^{5} b^{7} - 4 \, a^{3} b^{9} + a b^{11}\right )} d x \cos \left (d x + c\right ) + 24 \, {\left (a^{8} b^{4} - 4 \, a^{6} b^{6} + 6 \, a^{4} b^{8} - 4 \, a^{2} b^{10} + b^{12}\right )} d x - 3 \, {\left (20 \, a^{6} b^{5} - 35 \, a^{4} b^{7} + 28 \, a^{2} b^{9} - 8 \, b^{11} + {\left (20 \, a^{9} b^{2} - 35 \, a^{7} b^{4} + 28 \, a^{5} b^{6} - 8 \, a^{3} b^{8}\right )} \cos \left (d x + c\right )^{3} + 3 \, {\left (20 \, a^{8} b^{3} - 35 \, a^{6} b^{5} + 28 \, a^{4} b^{7} - 8 \, a^{2} b^{9}\right )} \cos \left (d x + c\right )^{2} + 3 \, {\left (20 \, a^{7} b^{4} - 35 \, a^{5} b^{6} + 28 \, a^{3} b^{8} - 8 \, a b^{10}\right )} \cos \left (d x + c\right )\right )} \sqrt {-a^{2} + b^{2}} \arctan \left (-\frac {\sqrt {-a^{2} + b^{2}} {\left (b \cos \left (d x + c\right ) + a\right )}}{{\left (a^{2} - b^{2}\right )} \sin \left (d x + c\right )}\right ) - {\left (6 \, a^{9} b^{3} - 71 \, a^{7} b^{5} + 133 \, a^{5} b^{7} - 92 \, a^{3} b^{9} + 24 \, a b^{11} + 6 \, {\left (a^{12} - 4 \, a^{10} b^{2} + 6 \, a^{8} b^{4} - 4 \, a^{6} b^{6} + a^{4} b^{8}\right )} \cos \left (d x + c\right )^{3} + {\left (18 \, a^{11} b - 132 \, a^{9} b^{3} + 239 \, a^{7} b^{5} - 169 \, a^{5} b^{7} + 44 \, a^{3} b^{9}\right )} \cos \left (d x + c\right )^{2} + 3 \, {\left (6 \, a^{10} b^{2} - 59 \, a^{8} b^{4} + 110 \, a^{6} b^{6} - 77 \, a^{4} b^{8} + 20 \, a^{2} b^{10}\right )} \cos \left (d x + c\right )\right )} \sin \left (d x + c\right )}{6 \, {\left ({\left (a^{16} - 4 \, a^{14} b^{2} + 6 \, a^{12} b^{4} - 4 \, a^{10} b^{6} + a^{8} b^{8}\right )} d \cos \left (d x + c\right )^{3} + 3 \, {\left (a^{15} b - 4 \, a^{13} b^{3} + 6 \, a^{11} b^{5} - 4 \, a^{9} b^{7} + a^{7} b^{9}\right )} d \cos \left (d x + c\right )^{2} + 3 \, {\left (a^{14} b^{2} - 4 \, a^{12} b^{4} + 6 \, a^{10} b^{6} - 4 \, a^{8} b^{8} + a^{6} b^{10}\right )} d \cos \left (d x + c\right ) + {\left (a^{13} b^{3} - 4 \, a^{11} b^{5} + 6 \, a^{9} b^{7} - 4 \, a^{7} b^{9} + a^{5} b^{11}\right )} d\right )}}\right ] \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\cos {\left (c + d x \right )}}{\left (a + b \sec {\left (c + d x \right )}\right )^{4}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.49, size = 564, normalized size = 1.89 \begin {gather*} -\frac {\frac {3 \, {\left (20 \, a^{6} b^{2} - 35 \, a^{4} b^{4} + 28 \, a^{2} b^{6} - 8 \, b^{8}\right )} {\left (\pi \left \lfloor \frac {d x + c}{2 \, \pi } + \frac {1}{2} \right \rfloor \mathrm {sgn}\left (2 \, a - 2 \, b\right ) + \arctan \left (\frac {a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )}{\sqrt {-a^{2} + b^{2}}}\right )\right )}}{{\left (a^{11} - 3 \, a^{9} b^{2} + 3 \, a^{7} b^{4} - a^{5} b^{6}\right )} \sqrt {-a^{2} + b^{2}}} - \frac {60 \, a^{6} b^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} - 105 \, a^{5} b^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} - 24 \, a^{4} b^{5} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 117 \, a^{3} b^{6} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} - 24 \, a^{2} b^{7} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} - 42 \, a b^{8} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 18 \, b^{9} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} - 120 \, a^{6} b^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 236 \, a^{4} b^{5} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 152 \, a^{2} b^{7} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 36 \, b^{9} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 60 \, a^{6} b^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 105 \, a^{5} b^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 24 \, a^{4} b^{5} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 117 \, a^{3} b^{6} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 24 \, a^{2} b^{7} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 42 \, a b^{8} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 18 \, b^{9} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )}{{\left (a^{10} - 3 \, a^{8} b^{2} + 3 \, a^{6} b^{4} - a^{4} b^{6}\right )} {\left (a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - a - b\right )}^{3}} + \frac {12 \, {\left (d x + c\right )} b}{a^{5}} - \frac {6 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )}{{\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 1\right )} a^{4}}}{3 \, d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 10.38, size = 2500, normalized size = 8.36 \begin {gather*} \text {Too large to display} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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